Gaussian Process for Interpreting Pulsed Eddy Current Signals forFerromagnetic Pipe Profiling

Nalika Ulapane∗, Alen Alempijevic∗, Teresa Vidal-Calleja∗, Jaime Valls Miro∗, Jeremy Rudd†, Martin Roubal†∗Centre for Autonomous Systems, University of Technology Sydney, Australia

† Rock Solid Group, Melbourne, Australia

Abstract— This paper describes a Gaussian Process basedmachine learning technique to estimate the remaining volumeof cast iron in ageing water pipes. The method utilizes timedomain signals produced by a commercially available pulsedEddy current sensor. Data produced by the sensor are usedto train a Gaussian Process model and perform inference ofthe remaining metal volume. The Gaussian Process model waslearned using sensor data obtained from cast iron calibrationplates of various thicknesses. Results produced by the GaussianProcess model were validated against the remaining wallthickness acquired using a high resolution laser scanner afterthe pipes were sandblasted to remove corrosion. The evaluationshows agreement between model outputs and ground truth. Thepaper concludes by discussing the implications or results andhow the proposed method can potentially advance the currenttechnological setup by facilitating real time pipe profiling.

Keywords-ferromagnetic, Gaussian process, machine learn-ing, non-destructive testing, pulsed Eddy current, sensor model

I. INTRODUCTION

Exposure of ferromagnetic water mains to aggressiveenvironmental conditions and deleterious reactions can leadto significant deterioration so as to undermine their abilityto function desirably and eventually fail through bursts [1].Therefore, the prediction of a pipe’s remaining life, espe-cially for critical water mains, is important for developingeffective renewal programs to manage pipe infrastructureand reducing the incidence of catastrophic failures. Betterunderstanding of the current condition and performance ofburied water mains and sewer pressure mains is an importantfirst step to help achieve improved understanding of remain-ing life. This is where the requirement for improved non-destructive metal pipe condition assessment techniques arise.

Techniques such as Pulsed Eddy Currents (PEC), RemoteField Eddy Currents, Magnetic Flux Leakage (MFL) andUltrasounds are commonly used for the purpose of exam-ining defects of ferromagnetic pipes [2]. Specifically, PECtechnology is being used for evaluating thickness of con-ducting material test pieces [3], [4], [5], corrosion detection[6], surface and subsurface defect detection [7] and pipeinspection [1], [6]. PEC technology usually characterises andquantifies defects, sensor liftoffs and material thicknessesby means of evaluating features extracted from the sensorresponse. Among the many possible features, the differencecurve features [4], [7], [8], logarithmic response features [9],[5], frequency spectrum features [10], unique characteristicsof sensor response [7], principal components [11] and inte-gral features [3] are the ones predominantly used. Certain

features extracted exhibit straightforward relationships toproperties of the test piece when the other properties arecontrolled. However, the problem is not that straight whenthe technology is used on cast iron pipes since there isminimal prior knowledge about the true nature of the testpieces and they are subjected to more complex contaminationin the form of graphitisation and surface and subsurfacedefects of arbitrary shapes and sizes.

Prior work on applying machine learning principles forEddy current profiling [12], [13] has not addressed thesophisticated problem of profiling the test piece where bothsurface and subsurface defects of arbitrary shapes are present.For instance [12] classifies cracks based on the straight crackhypothesis and the proposed method is limited when tryingto address arbitrary shaped cracks. However, as mentionedin [12], more general problems may be solved by increasingthe evaluated parameters in the training data set. The form ofdefects present in aged cast iron pipes is not limited. Thus,existing feature extraction techniques are not transferable toprofiling cast iron pipes with complex geometry, the fullsensor response instead of few features for training could beutilised. As cast iron material is ferromagnetic, this resultsin weak penetration of Eddy currents because of the skineffect and further exacerbates the need to exploit the fullresponse that can be obtained. All work related to PECsignal interpretation found in literature, even the ones usingmachine learning [12], [13], have been attempted on non-ferromagnetic materials. However, [14] is one exceptionwhere PEC technology has been used in conjunction withRemote Field Eddy Currents for testing ferromagnetic tubes.

This paper brings novelty in using the Bayesian estimationframework incorporating Gaussian processes (GPs) [15] toestimate the remaining volume of cast iron. GPs are non-parametric models that can be used to solve non-linearregression problems such as the one in hand. GPs havebeen recently applied in this context to identify defects insteel plates with MLF sensors [16]. In this paper, PECsensor signals are used as input for this non-parametricmodel. The GP model is trained using cast iron calibrationplates of various thicknesses. Thereafter the trained modelis used to estimate the remaining wall thickness of in-situpipes. Validation of the estimated results is performed onthe pipes once they are sandblasted to remove corrosion.The evaluation is performed on two 1 m sections of a castiron pipe with 600 mm external diameter showing agreementbetween model outputs and ground truth with 3.08 and

1.71 mm root mean square error and standard deviationrespectively.

The paper unfolds under four sections from here onwards.First the theory behind the sensor architecture and the GPsapproach are explained in section II. It is followed by sectionIII which explains the experimental work carried out alongwith the procedures of training, testing and validating the GPmodel. Section IV includes graphical and numerical resultswhich exhibit the validity of the GPs outputs. Finally, theconcluding remarks are included in section V along withthe implication of results and potential improvements to theproposed method.

II. APPROACH

The proposed approach uses supervised learning to traina non-parametric model, which gets input features extractedfrom the PEC sensor signals and produces an estimate ofthe remaining wall thickness of the ferromagnetic materialunderneath the sensor. As can be appreciated from the afore-mentioned, the ability to estimate remaining wall thicknesscomprises of three primary components. Firstly, a sensorarchitecture capable of energising and measuring the Eddycurrents in a ferromagnetic material, following a subsampling(feature extraction) step and a machine learning framework,all of which are detailed in the following sub-sections.

A. Sensor principles

In a typical PEC sensor, the primary coil is excited witha voltage pulse. This pulse creates a time varying magneticfield around the sensor, the resulting magnetic field inducesEddy currents in the conducting test piece (i.e. cast iron pipe)as shown in Fig. 1. Induced Eddy currents create a magneticfield, which opposes the field generated by the exciter coil.The resultant time varying magnetic field induces a voltage inthe detection coil. This induced voltage is the output signalproduced by the sensor and it takes the shape of a decaycurve as shown in Fig. 2. The time dependant decay curvesuniquely characterise properties of the test piece. The outputsignal (induced voltage) produced by the antenna are filtered,digitally processed and recorded. Fig. 3 shows a basic blockdiagram of the macro view of the full sensor setup. However,details about the dimensions of the antenna, amplitude andfrequency of the sensor excitation sequence and the internalelectronics are omitted. The approach presented herewith isgeneric in nature and does not critically depend on theseparameters.

The fundamental principle of the sensor, as describedin [17], is as follows. If the resultant magnetic vectorpotential at an arbitrary location inside the domain of thedetection coil is expressed as A, the coil current density Jat that location is given by

52A− µσdAdt

= −µJ , (1)

where µ and σ are the permeability and conductivity of thedetection coil material.

When the current density at a point is known, the electricfield intensity E at that point is given by J = σE and

Fig. 1. Cross sectional view of the Comsol Multiphysics R© model showingthe Eddy current induction phenomenon.

Fig. 2. Induced voltage of the detection coil known as decay curve.

Fig. 3. Diagram of the full sensor setup.

therefore, it is known that the detection voltage V can becalculated by performing the line integral

V =

∫C

Edl (2)

along the total length of the detection coil.To model any type of Eddy current sensor requires solving

(1) and (2) and this work is no exception. Thus, the sensorresponse shown in Fig. 2 was generated by solving (1) and(2). To illustrate the anticipated sensor response, a solution isprovided using Finite Element Analysis (FEA) [17] with theaid of the commercial FEA package Comsol Multiphysics R©.Fig. 4 shows the complete 3D model of the PEC sensorplaced above the cast iron pipe, made in the FEA package.The phenomenon of Eddy current induction in the pipe canbe visualized in the cross sectional image in Fig. 1. It is clearfrom Fig. 1 that the domain of influence (the area in whichthe Eddy currents are induced) is much larger than the areaphysically covered by the sensor.

B. Feature Extraction

Feature extraction is a well-known mechanism to reducethe dimensionality of the data. It maps raw sensor signalsinto a lower dimensional space. Features, in this case, arederived from full time-varying voltage signals produced by

Fig. 4. FEA simulation model of the sensor placed on a cast iron pipe,made in Comsol Multiphysics R©

the sensor. The voltage decay curve obtained by the sensor,shown in Fig. 2 is subsampled with a logarithmic timescalesimilar to that in [18]. Thus, a subsampled version of thedecay curve is produced with a very high sampling rate closerto the beginning of the decaying section of the curve at 1msand the last sampled instance being equal at the total signalduration of 58ms. A total of 24 subsamples were acquiredin this interval as illustrated in Fig. 5 and used as featuresin the subsequent machine learning approach.

C. Gaussian Process Formulation

Estimating thickness from PEC sensor signals can beformulated as a non-linear regression problem. GaussianProcess models are a powerful tool to solve such regressionproblems. GPs can be thought of as a Gaussian prior overthe function space mapping inputs x and outputs f(x). It iscompletely specified by its mean function µ = E[f(x)] andthe covariance function Σ = E[(f(x)−µ)(f(x)>−µ>)] [15].

Let [X,Y ] be the training data set drawn by the noisyprocess yi = f(xi) + ε, where X = [x1, x2, ..., xm]T

be a matrix of training inputs that in our particular casecorresponds to the features extracted from the PEC signal andY = [y1, y2, ..., ym]T be a vector of training labels, whichare the corresponding cast-iron plates thickness. A GaussianProcess estimates posterior distributions over functions ffrom the training data [X,Y ]. Although the functions areinfinitely dimensional, the GP model is used to infer, or

Fig. 5. Subsampled logarithmic timescale decay curves.

predict, function values at a finite testing set of predictionpoints X∗ = [x∗1, x

∗2, ..., x

∗n]T .

To apply a GP framework to this regression problem, onemust first select a kernel K(X,X) whose elements are givenby ki,j = k(xi, xj). This specifies the kind of functions thatare expected, before any data have been seen. Technically,the kernel places a prior likelihood on all possible functions.After evaluating a number of commonly used kernels, thesquared exponential kernel has been chosen for this work. Itis defined as

k(xi, xj) = α2 exp

{− 1

2β2(xi − xj)2

}, (3)

where α and β represent its hyper-parameters and togetherwith sensor noise variance σn are learned from the trainingdata. The GP model is trained by minimizing the negativelog marginal likelihood in (4) with respect to θ = {α, β, σn}.

− log p(Y |X, θ) =1

2Y T Σ−1Y +

1

2log |Σ|+ m

2log(2π) ,

(4)where the covariance function is given by

Σ = K(X,X) + σ2nI . (5)

The combination of the training data and the kernelinduces not only the most likely state, but also a full posteriorprobability distribution. The basic GP regression equationsare given by

µ∗ = K(X∗, X){K(X,X) + σ2nI}−1y (6)

Σ∗ = K(X∗, X∗) + σ2nI −K(X∗, X)

{K(X,X) + σ2nI}−1K(X,X∗) ,

(7)

where I is the corresponding identity matrix.The expected pipe wall thickness for the testing input

vector X∗ will be therefore given by the mean of theposterior distribution µ∗ and the associated uncertainty willbe given by the covariance Σ∗.

III. METHODOLOGY

The complete proposed methodology is presented inFig. 6. Three main tasks were carried out in this work: train-ing the GP model, generating the pipe profiles and validatingthe model outputs. The methods to accomplish these tasksare explained in detail in the following subsections.

A. Training the GP Model

Training data were obtained by scanning cast iron cal-ibration blocks of thicknesses 1 through 10 and then12, 14, 16, 18, 20, 22, 25, 30 and 35 mm, as generally done bythe PEC technology provider. Note that the more data usedfor training the less uncertainty associated to the output ofthe GP model. The training data for this work was obtainedby maintaining the sensor liftoff constant at 0 mm. Thisreplicates the physical gap that will exist between the sensorand the pipe when obtaining readings in-situ as in Fig. 7.

Fig. 6. Complete methodology of training the GP model, generating pipeprofiles and validating.

As mentioned previously in section II-B, the subsampledlogarithmic time domain decay curves were used as featurevectors for this work to overcome the restrictions associatedwith standard features extracted from the curves. The setof subsampled decay curves together with the associatedthickness were used as to learn the GP model.

An example of the posterior drawn by the GP inferencefor two selected features is shown in Fig. 8. The non-linearfunctions associated with these features and captured bythe GP model show complimentary discriminative powers.While feature 10 discriminates well in the region between8-16 mm thickness as illustrated in Fig. 8(a), feature 18holds discriminative power in the 14-25 mm region as perFig. 8(b). Additionally, higher uncertainty in the GP modelis observed in regions where insufficient training data isprovided. For example, an inference of 13 mm thickness hasdemonstrably high uncertainty, whereas inferences of 12 mmand 14 mm where training data is provided have substantiallylower uncertainty bounds as denoted in Fig. 8(a). Finally, itcan also be ascertained that neither feature in Fig. 8 hassubstantial discriminative power in thickness above 25 mm.This is caused by a combination of insufficient training dataand reduced eddy currents being generated further away fromthe exciter, both factors will be addressed in future work.

Fig. 7. Obtaining testing data (BEM scanning being done on the test bed).

(a) Feature 10

(b) Feature 18

Fig. 8. GP inference of thickness for two features from the consideredfeature vector

B. Obtaining Testing Data

Testing data were obtained by scanning the full circumfer-ence of pipe segments of 1 m length along the pipe axis onthe test bed allocated for this work. Reference [2] providesdetails about the 1.5 km test bed, two 1 m pipe segments wereidentified by Sydney Water as possibly defective and requiredmore in depth inspection. Fig. 7 shows how BEM scanningis done on the pipe test bed. To account for variability inelectro-magnetic properties of cast iron the training data andtesting data from each pipe segment are jointly normalisedand standardised. As estimation of thickness is not performedin real-time, this limitation will be addressed in future work.

C. Generating 2D Pipe Profiles

Once the GP model is trained, the testing decay curvesobtained from the test bed are used to infer the pipe profilesas 2D thickness plots along with the uncertainty. Featuresare extracted as described in section II-B and, (6) and (7)are used to produce the estimated thickness.

D. Validating the GP Model Outputs

The test bed pipe segments were exhumed and bothsurfaces (external and internal) were sandblasted to strip offany rust or graphitization in order to obtain segments withclean metal [2]. The external and internal surfaces of thesandblasted pipe segments were scanned using a 3D laserscanner and accurate 3D pipe profiles were generated usinga ray tracing algorithm described in [2]. Shown in Fig. 9is an example of the obtained 3D laser profile of a scannedpipe segment.

Note that the resolution of the laser profiles is 2 mm,while that of the PEC sensor is 5 cm. Therefore, the 3Dprofile thicknesses were averaged to obtain the 2D thicknessplots for qualitative and quantitative comparison. The GP

inference results were compared with the ground truths toevaluate the accuracy as presented in the following section.

IV. RESULTS AND DISCUSSION

This section includes the results produced by the GP modeland the comparison against the ground truth. As mentionedabove results of two pipe segments are included.

Fig. 10 and 11 show the GP inferred thickness plotsand corresponding uncertainties of the two segments. Thethickness and uncertainty plots represent the profile of a660mm diameter full circumferential pipe segment of 1000mm in length along the x direction which is parallel to theaxis of the pipe. A comparison of the inferred profiles againstthe ground truth and the profiles produced by the technologyprovider (RSG) are shown in Fig. 12 and 13. The whitesections in the GP plots are sensor readings that were ignoreddue to anomalies caused by not having the hand held BEMsensor stable for a sufficient duration when acquiring datafor that section of the pipe.

A quantitative comparison is presented in Table I andII showing that the results produced by the proposed GPapproach are marginally better (closer to the ground truth)than the results produced by RSG. The results show thatfor both segments, the RSG and our GP inference, are inaccordance with the ground truth. The proposed approach,however, produces uncertainty plots that give a measure ofthe reliability on the estimated thickness as shown on theright hand side of Figs. 10 and 11. The uncertainty variesdepending on how close the estimated thickness is to thetraining data, the more training samples the more accurateresults. Fig. 14 highlights a region where the uncertainty is

Fig. 9. 3D laser profile of the scanned pipe (Ground Truth).

Fig. 10. Interpreted pipe profile and uncertainty of segment 1.

Fig. 11. Interpreted pipe profile and uncertainty of segment 2.

TABLE ICOMPARISON STATISTICS OF SEGMENT 1, THE ERROR BETWEEN

INTERPRETED THICKNESS AND GROUND TRUTH

Value GP RSG

Mean 2.71 mm 2.96 mmStandard Deviation 1.76 mm 1.50 mmRoot Mean Square 3.23 mm 3.29 mmMinimum 0.02 mm 0 mmMaximum 10.4 mm 7.28 mm

low when estimating areas with lower thicknesses, implyinghigher reliability. The underlying reason is the larger amountof training data available for lower thicknesses.

V. CONCLUSIONS

This paper proposed a GP based machine learning tech-nique for interpreting PEC signals for profiling the remainingwall thickness in ferromagnetic cast iron water pipes. Theapproach utilises the full sensor response, subsampling theoriginal signal on a logarithmic scale to produce a featurevector which is different from the specific feature basedwork available in literature. The Gaussian Process model

Fig. 12. Pipe segment 1(Comparison against Ground Truth).

Fig. 13. Pipe segment 2(Comparison against Ground Truth).

TABLE IICOMPARISON STATISTICS OF SEGMENT 2, THE ERROR BETWEEN

INTERPRETED THICKNESS AND GROUND TRUTH

Value GP RSG

Mean 2.41 mm 2.48 mmStandard Deviation 1.67 mm 1.76 mmRoot Mean Square 2.93 mm 3.03 mmMinimum error 0 mm 0 mmMaximum error 8.14 mm 7.90 mm

Fig. 14. Zoomed view of Pipe segment 1 and associated Uncertainty.

was trained using sensor signals obtained from cast ironcalibration plates of various thicknesses. Results producedby the Gaussian Process model were validated against theremaining wall thickness with agreement observed betweenmodel outputs and ground truth (root mean square error 3.08and standard deviation 1.71 mm). Profiling in this work iscoarse due to the constraints and design of the sensor used.

Future work considers fusing several PEC sensor mea-surements in overlapping configurations to produce higherresolution pipe profiles and altering the excitation waveformto enable discrimination of larger wall thicknesses. There isalso room for designing more sophisticated PEC sensor ar-chitectures to overcome obstacles like skin effect for accurateprofiling of ferromagnetic materials.

ACKNOWLEDGMENT

This publication is an outcome from the Critical PipesProject funded by Sydney Water Corporation, Water Re-search Foundation of the USA, Melbourne Water, WaterCorporation (WA), UK Water Industry Research Ltd, SouthAustralia Water Corporation, South East Water, Hunter Water

Corporation, City West Water, Monash University, Universityof Technology Sydney and University of Newcastle. Theresearch partners are Monash University (lead), Universityof Technology Sydney and University of Newcastle.

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